# Weighted estimates for the bilinear maximal operator on filtered measure spaces

## TL;DR

This paper establishes weighted inequalities for the bilinear maximal operator on filtered measure spaces under the bilinear reverse Hölder condition, introducing new techniques like bilinear principal sets and a Carleson embedding theorem.

## Contribution

It introduces a novel construction of bilinear principal sets and a new Carleson embedding theorem tailored for filtered measure spaces, advancing weighted inequality theory.

## Key findings

- Weighted inequalities are established under the bilinear reverse Hölder condition.
- Hytonen-Perez type weighted estimates are obtained for the bilinear maximal operator.
- A new property called conditional sparsity of principal sets is identified.

## Abstract

Assuming the bilinear reverse Holder's condition, we character weighted inequalities for the bilinear maximal operator on filtered measure spaces. We also obtain Hytonen-Perez type weighted estimates for the bilinear maximal operator. Our approaches are mainly based on the new construction of bilinear versions of principal sets and the new Carleson embedding theorem on filtered measure spaces. In particular, we find a new property of the construction and we call it the conditional sparsity of principal sets.